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We present an efficient matrix-free geometric multigrid method for theelastic Helmholtz equation, and a suitable discretization. Many discretizationmethods had been considered in the literature for the Helmholtz equations, aswell as many solvers and preconditioners, some of which are adapted for theelastic version of the equation. However, there is very little work consideringthe reciprocity of discretization and a solver. In this work, we aim to bridgethis gap. By choosing an appropriate stencil for re-discretization of theequation on the coarse grid, we develop a multigrid method that can be easilyimplemented as matrix-free, relying on stencils rather than sparse matrices.This is crucial for efficient implementation on modern hardware. Using two-gridlocal Fourier analysis, we validate the compatibility of our discretizationwith our solver, and tune a choice of weights for the stencil for which theconvergence rate of the multigrid cycle is optimal. It results in a scalablemultigrid preconditioner that can tackle large real-world 3D scenarios.

LFA-tuned matrix-free multigrid method for the elastic Helmholtz equation